How do i factor this

Factor the trinomial.

20x^2 - 43x + 14


Factor the problem completely. First, factor out the greatest common factor, and then factor the remaining trinomial.

3y^4 - 3y^3 - 168y^2

(5x - 2) (4x - 7)
3y^2 (y-8)(y+7)

You are looking for 4 numbers.

Two of them will multiply out to +20
The two others will multiply out to +14
Their cross products (one number from one pair times a number from the other pair), will add up to -43

This means that at least one pair must have negative numbers (otherwise, can't add up to -43).

Let's try with

20 = 4*5
14 = -7 * -2

cross product: (4 * -2) + (5 * -7) = -8 + -35 = -43
OK, we have our numbers.

pair the elements in the order NOT used in the cross product:
4 with -7
5 with -2

(4x - 7)(5x - 2)

expand this product (as a verification)

(4x - 7)(5x - 2) = 4x(5x - 2) - 7(5x - 2)
= 20x^2 - 8x - 35x + 14 = 20x^2 - 43x + 14
Yep! it works.

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The obvious common factor is y^2

You can do this many ways, but the formal, detailed method goes like this:
if you multiply outside the bracket by a number, then you must divide the inside of the bracket by the same number:

(3y^4 - 3y^3 - 168y^2) =
y^2( 3y^4/y^2 - 3y^3/y^2 - 168y^2/y^2) =
y^2(3y^2 - 3y - 168)

Most of us simply do it in one shot, skipping the part about writing down the individual divisions.
Next, we notice that the coefficients are all divisible by 3, so we factor out the 3 as well:

3y^2(y^2 - y - 56)

You can continue factoring the inside of the bracket. You can use the same method as above

you need two numbers that multiply out to +1
(usually, we stick with 1 and 1)
you need two numbers that multiply out to -56 (one number must be negative)
AND the cross product adds up to -1
From that, we try to find two numbers that follow each other (difference of 1) that will multiply out to 56

How about 7 and 8

And the negative one will be the larger value (8) because the sum is negative.

(1y + 7)(1y - 8) = y^2 + 7y - 8y -56 = y^2 - y -56
(in real life, we don't bother writing the "1" as a coefficient).

Now, all is left is to put everything together.

Use the quadratic formula, then
[x - root1][x - root2]

For the second one, strain out y^2 and you have a quadratic left; factor that per above